The Grimoire is a textbook on commutative algebra
for advanced master students and graduate students that
I am currently writing. Some of its last chapters could
be interesting also to more advanced researchers. It is
written in French. Here you can download the current version :

Your comments are welcome and gratefully acknowledged!

Status Update :

November 8, 2014 : work on the Grimoire project
has resumed. Lectures
Bélier, Taureau et Gémeaux are complete; currently
working on Cancer.

December 7, 2014 : Lectures
Bélier, Taureau, Gémeaux et Cancer are complete;
currently working on Lion.

March 6, 2015 : Lectures
Bélier, Taureau, Gémeaux, Cancer and Lion are complete;
currently working on Vierge.

May 23, 2015 : The Grimoire Project enters Summer recess : until
the end of September I will be working on a different project,
so there will be not much action for the Grimoire : I might make
the occasional correction, and maybe toy with the index once and
then, but other than that don't expect much until October.
The Vierge lecture is not quite
finished yet, but it is well advanced.

October 14, 2015 : work on the Grimoire project has resumed.
Currently working on Vierge.

October 23, 2015 : Lectures
Bélier, Taureau, Gémeaux, Cancer, Lion et Vierge are complete;
currently working on Balance.

November 1, 2015 : I've corrected the proof of Zariski's Main Theorem,
that was confusing and slightly broken in various places. Please
download the revised version, and keep on sending me your remarks.
I've also added a result on UFDs with some applications, at the end
of section 11.4. Now, back to Balance.

December 15, 2015 : Lectures
Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge and Balance
are complete; currently working on Scorpion.
I've also made some small corrections to the proof of Swan's theorem
in Lecture 5 (Lion), and I've added some
exercices at the end of section 6.5.

February 1, 2016 : I've taken the important decision to add
a section (a revised section 5.5 that I'm writing now) on the
language of sheaves and a few basic notions of scheme theory.
This has entailed some reorganization in lectures
Lion, Vierge and Balance.
The reason to include this material is that eventually I'm planning
to give a presentation of a valuation-theoretic proof due (essentially)
to Fujiwara of Gruson-Raynaud's important theorem on flattening of
morphisms by admissible blow-ups. This proof is more elementary
than Gruson and Raynaud's original argument, but of course to
properly state the theorem and to give a clean presentation
of its proof it is convenient to be able to manipulate some
non-affine schemes (such as blow-ups).

April 7, 2016 : I've finally completed the digression on
sheaves and schemes announced in February; it is longer
than originally planned and is now split over two
sections : the new section 3.4 on sheaves and the new
section 5.5 on schemes. This entailed some rearranging
of material around the text. Now I'm ready to return to
Scorpion.

April 29, 2016 : The Grimoire Project enters Summer recess : until
late September I will be working on a different project, so not
much will happen on this page. Scorpion
is not quite finished, but is well advanced. See you in a few months!

November 21, 2016 : The Grimoire Project is active again!
I was very busy until now, but at last I am free. I've just
started a new section in Scorpion,
dedicated to the real spectrum of a ring : it doesn't contain
much at present, but watch it grow!

January 16, 2017 : Lectures
Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance and
Scorpion are complete; next up is
Sagittaire.

April 13, 2017 : Lectures
Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance,
Scorpion and Sagittaire are complete. I've also
made several minor fixes in earlier parts of the text, and
added several new entries to the index.
There will be a few further updates next week, then
the Grimoire Project will enter Summer recess.

December 12, 2017 : work on the Grimoire Project has
finally resumed! Currently working on
Capricorne.

March 4, 2018 : today the Grimoire Project has reached
an important milestone : the proofs of both the Gruson-Raynaud
theorem and of elimination of quantifiers for fields with
non trivial valuations are complete : see theorems 10.57
and 10.62 (I give a geometric form of elimination of
quantifiers : see the remarks before paragraph 10.4.3).
Still working on Capricorne :
section 4 is almost done, and it remains (only) to write section 5.

Mai 5, 2018 : Lectures
Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance,
Scorpion, Sagittaire and Capricorne are complete. Today the
Grimoire also enters officially into summer recess. I
might do some little cleaning up, and also add a few entries
to the index, but that will be it until October at the
earliest, I expect.

June 27, 2018 : I took advantage of some free time to make a
thorough reorganization of the last 4 chapters of the Grimoire
(and I've also added an exercise in what is now section 9.2).
The upshot is that, for the time being, the Grimoire is complete
only up to Sagittaire, so in a sense
this is a step backward, but it's only in order to prepare a leap
forward that I'll take after the summer : then hopefully I'll be
able to complete both Capricorne and
Verseau in a relatively short time.
Also, you can start seeing now the general shape that
Poissons will have in the end : more
material is planned, but the basic scaffolding is in place.

September 23, 2018 : work on the Grimoire Project has resumed!
Currently working on the revised version of
Capricorne.

November 2, 2018 : I'm finally done with the revision of
Capricorne! It took slightly
longer and slightly more effort than I had anticipated,
because while working on it, I realized that the previous draft
was rather (and embarassingly) rough : now it is much better.
Hence : lectures
Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance,
Scorpion, Sagittaire and Capricorne are again complete,
and I'll start working on Verseau
in a matter of days, after I rest a little bit with a short break.

December 30, 2018 : During the holidays I'm not working on
Verseau, but I'm not quite idle
either : I just finished fixing a very old and embarrassing
mistake : the homological and cohomological notations for
complexes were all mixed up, throughout the Grimoire. I have
been aware of the problem for years, but I had never found the
time and the nerve to correct this : now it's done. I'll resume
working on Verseau around the
second week of January : stay tuned!

January 26, 2019 : Lectures
Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance,
Scorpion, Sagittaire, Capricorne and Verseau are
complete. I'm taking a well deserved break from the Grimoire,
to take care of a bunch of long-delayed other smaller projects.

April 22, 2019 : I've just completed a major revision of
Bélier : two of the previous
sections (1.3 and 1.4) have fused into a single longer
one, and a new section 1.4 has been added, containing a
self-contained treatment of the elementary theory of fields;
moreover, some material on unique factorisation domains has
been moved from section 5.2 to section 1.1.

May 8, 2019 : today the Grimoire project enters officially
into summer recess : I might add some entries to the Index,
but do not expect any substantial work until October at the
earliest. The last addition to the text so far is problem
12.27, detailing Nagata's example of an infinite dimensional
noetherian domain. See you in a few months!

September 29, 2019 : work on the Grimoire Project has resumed!
Currently working on Poissons.
The first new result is a proof of the classical Molien formula
for the Hilbert-Poincaré series of the ring of polynomial
invariants for the action of a finite group on a finite-dimensional
vector space : see Problem 12.55.

January 4, 2020 : Today the Grimoire reaches an important
milestone : lectures
Bélier, Taureau, Gémeaux, Cancer, Lion, Vierge, Balance,
Scorpion, Sagittaire, Capricorne, Verseau and Poissons are
complete : that's the entire Zodiac! However, this is not yet
the end of the project, since I'll still make some substantial
revision and improvements to some earlier chapters : stay tuned!

February 21, 2020 : Today I've completed the first of a planned
series of major revisions of the Grimoire : the objective of
this revision was to include the proof of a classical result,
stating that one can desingularize an integral noetherian scheme
of dimension one (that is, a curve) by a canonical sequence of
blowing-up morphisms centered at the singular loci, provided that
the normalization map of the curve is a finite morphism (this is
really the minimal condition required for the existence of such a
desingularizing sequence of blowing-ups). This theorem is now
contained in section 7.4 : see theorem 7.58. The proof in itself
is not so long, but it uses a good part of the material developed
in the first 7 lectures, including Zariski's Main Theorem; I had
to add a certain amount of preliminaries concerning schemes, and
that's why sections 5.4 and 5.5 now are considerably longer and
look rather different, and sections 5.2 and 5.3 have fused into a
single section. Moreover, some scheme-theoretic foundational
material that was previously scattered in lectures 10 and 11 now
has been also moved and reorganized in these two sections 5.4 and 5.5.

April 19, 2020 : The second major revision of the Grimoire
is now essentially complete : this revision has invested
and completely overhauled the last three lectures. So,
the material that formerly occupied
Poissons now has become
Verseau, and the bulk of what
used to be Verseau and
Capricorne has been reorganized
between Capricorne and
Poissons, with many notable
additions, and some deletion, as the new material has
superseded previous treatment in some section. Moreover,
section 5 of Sagittaire
is now devoted to the Artin-Rees lemma (formerly a topic
in Verseau), and the material
concerning Huber rings is now collected in section 4.
What used to be section 5 of Sagittaire,
(on continuous valuations) has become the first section of
Capricorne, and this lecture
now ends with a section 5 dedicated to the Gruson-Raynaud
theorem. The generalities on affinoid domains and adic spaces
now occupy the first section of Poissons;
the second section is completely new, and presents the
theory of analytically noetherian rings, superseding the
previous treatment of strongly noetherian rings (which
are now a special case); the advantage is that analytically
noetherian rings also encompass the Huber rings with a
noetherian ring of definition, which were treated by Huber
separately, with methods from algebraic geometry. Instead,
we derive all the basic features of (universally) analytically
noetherian rings -- including strict acyclicity of the Cech
complexes with coefficients in the structure presheaf -- in
a unified fashion, by an essentially combinatorial analysis
that generalizes the original treatment of Tate. I might add
some further complements later on, but essentially we're done
with this second major revision : only one major revision
left before we're done!

May 9, 2020 : I am currently halfway through the last major
revision. The aim of this revision is to include some material
concerning abelian categories, up to and including the
Freyd-Mitchell's theorem. So far I've added the basics
of abelian categories, all the way to the snake lemma.
This has entailed a thorough reorganization of
Bélier and Taureau. The changes
in Bélier are relatively minor,
but Taureau is now completely
different, and is wholly devoted to category theory. Right
now Gémeaux is in a messy state,
and at the end of the revision it will look very different;
the same goes for Cancer and Lion
which will be both seriously affected by this revision.
Another important change is that I've spent some efforts
in improving the treatment of certain set-theoretic issues
having to do with the manipulation of large categories;
now our set-theoretic framework is explicitly spelled out
(the Grimoire officially adopts the Bernays-Goedel axiomatics
for set theoretic foundations), and a certain care is applied
in distinguishing between small and large categories,
especially in the treatment of limits and the constructions
of categories of functors. Until now, these issues could be
just swept under the rug, but the proof of the Freyd-Mitchell's
theorem involves handling some very large constructions, so
the whole proof gets very shaky if one neglects dealing
with these size issues.

May 29, 2020 : The final major revision of the Grimoire is not finished
yet, but as of today Gémeaux has again
reached a stable configuration, and it will no longer change. This
chapter now contains the material on tensor products and on localization
of rings (including Nakayama's lemma); the main change is that now I
introduce tensor products from the point of view of bimodules : this
is the only reasonable way to proceed if one wants to include the case
of modules over non-commutative rings, but has also the benefit of
streamlining the treatment in several places, even in the commutative
case. The non-commutative case has been gradually forced on me, because
of the Freyd-Mitchell theorem, which embeds every abelian category fully
faithfully and exactly into a category of modules over some (usually
non-commutative) ring.
Anyway, whereas Gémeaux is now stable,
both Cancer and
Lion are still in flux, though they are,
one could say, structurally sound : for Cancer,
only section 1 is still incomplete : this is dedicated to advanced
material on abelian categories; so far it contains a complete proof
of the Gabriel-Popescu theorem (my proof is somewhat simpler than
the one that is found in Kashiwara-Schapira's book on
"Categories and Sheaves", but I don't know if it is new : for all
I know, it might be the original proof of Gabriel and/or Popescu)
and also of Grothendieck's classical theorem on the existence of
injective envelopes in Grothendieck categories (aka cocomplete
well-powered AB5 categories); my proof here is essentially due to
Mitchell, though I reconstructed it from a very incomplete and sloppy
account of it found in a little book by Freyd : what I like about it,
is that it doesn't require transfinite induction : Zorn's lemma
suffices (well, it always does... but the switch from one way of
writing things to another is not always easy or natural... and this
proof is not just some transcription of Grothendieck's proof : it's
really a different proof).
What is missing from this section is the Freyd-Mitchell theorem, and
then we will be done! Likewise, in Lion,
only section 1 will change : it's about homotopies and resolutions,
but so far it's only for chain complexes of modules, and I will do it
for chain complexes in any abelian category; the same goes for sections
7.5 and 8.5. But for now I'm taking a much deserved break.

July 22, 2020 : As of today, Cancer
is again stable, and will no longer change. So, section 1 of this
chapter is now complete, and ends with the already announced proof
of the Freyd-Mitchell theorem. The next chore is to revise section
1 of Lion, dedicated to resolutions
and homotopies of complexes : currently it deals only with chain
complexes of modules over a given ring, and it will be upgraded
to general abelian categories. Similarly, I will need also to
upgrade sections 7.5 and 8.5.

July 28, 2020 : As of today, also Lion
is again stable, and will no longer change. Only section 1 needed
revising, and I also updated the introduction to this chapter.
Next up would be sections 7.5 and 8.5, but I'll probably work
on them a little later.

August 10, 2020 : Another important milestone for the Grimoire : as
of today, the Index is complete : it is
15 page long (14 pages for the online version that you can download)
and it lists 769 items!

August 13, 2020 : Today I've completed the revision of section 7.5.
This is the section dedicated to the construction of derived functors
of additive functors; previously it was only for functors between
categories of modules, and now it deals with additive functors
between general abelian categories : olé!

August 21, 2020 : Today I've completed the revision of section 8.5,
dedicated to double complexes. Previously it only dealt with double
complexes of modules, and now it covers double complexes in any
abelian category. This achieves the last major revision of the
Grimoire. There remain a few small additions and minor improvements
to finish up, but the end is nigh!

October 12, 2020 : Today I've completed a thorough overhaul of
the treatment of faithfully flat descent (see pages 303--310).
The new version is a combination of conceptual arguments and
explicit calculations with tensors; on the "conceptual" side,
the main novelty is the introduction of cartesian functors (in
the special situation that is needed for faithfully flat descent,
so I do not develop a fully-fledged theory of fibrations).
The idea of cartesian functors improves the foundations of descent
theory, but is also applied later, for the proof that the category
of quasi-coherent modules on Spec(A) is equivalent to that of
A-modules (problem 5.57(i.b)). For this, I already used faithfully
flat descent early on, but the previous treatment was very awkward,
since the lack of adequate foundations forced me to proceed by
checking painstakingly the commutativity of several annoying diagrams
of modules... This was the part of the Grimoire which I considered
the least satisfactory; now the proof looks completely different,
and is very much improved.

November 29, 2020 : Today I've finally completed the last important
addition to the Grimoire : it's a generalization of the work of
Buzzard and Verberkmoes which establishes that so-called stably
uniform Tate-Huber rings are "sheafy" : the version of the Grimoire
considers more generally arbitrary Huber rings (a Tate-Huber
ring is a Huber ring that has a topologically nilpotent unit).
One interesting aspect is that the proof of sheafiness in this
case is achieved by the same method that the Grimoire uses for
proving "sheafiness" (aka Tate acyclicity) in all the other
cases : so now we have a single method for showing Tate
acyclicity that works uniformly in all the currently known
cases, and is essentially elementary(!)
Especially, this includes the case of perfectoid rings! That's
because perfectoids are stably uniform : this was the original
motivation of Buzzard & Verberkmoes. Moreover, the version of
the Grimoire applies even to the more general perfectoids that
Gabber and me introduce in our chunky manuscript on
"Foundations for Almost Ring Theory" : olé!

December 12, 2020 : As of today, the Grimoire Project is complete!
Rejoice! Shout it from the rooftops!
Now comes the last phase : the final reading of the whole manuscript,
which I expect will last at least 6 more months : so, you're still
in time to send me your observations/corrections/encouragements.
The final reading will start in a week or so, after a short and
well deserved break.
Thank you very much for accompanying me in this adventure!

January 8, 2021 : The final reading of Bélier
is now complete! I corrected zillions of misprints and minor (sometimes,
not so minor) mistakes, and made a good number of tiny improvements :
now is as perfect and sublime as it will ever get!
Only 11 chapters to go!

February 19, 2021 : The final reading of Taureau
is now complete! Besides correcting countless misprints and minor
mistakes, the most notable change is the addition of the construction
of the quotient of an abelian category by a Serre subcategory : this
is now Problem 2.101. I tried to be careful with the set-theoretic
issues involved in this construction : one needs that the ambient
category is well-powered; then, even with this assumption, one
has to be careful in the construction of the sets of morphisms in
the quotient category; this makes some arguments slightly less
transparent than how one might want, but that seems to be the price
to be paid in order to achieve a fully rigorous proof.

March, 2, 2021 : The final reading of Gémeaux
is now complete : halleluyah!! I corrected tons of misprints and minor
mistakes, as usual, but no substantial changes were required, so this
was a quickie : only two weeks for this chapter. Onward to
Cancer!

May 5, 2021 : The final reading of Cancer
is now complete! This took much longer : almost exactly two months.
That's partly due to all the teaching I had to do this semester, which
slowed me down considerably; but it's also because this was a
complex and long chapter to read and revise. Apart from the usual
few tons of corrections of minor mistakes, the main changes are
in section 4.3, dedicated to presheaves and sheaves. Especially,
I've decided to redo some of the basic constructions for sheaves
and presheaves using the language of Kan extensions : then, I had
to revise also parts of Taureau, to
include basic material on Kan extension (which I've added to section 2.3).
Another addition is exercise 4.88(ii), proving that the presheaves
and sheaves of modules (over any given ring) form a Grothendieck
abelian category with a generator; in particular, this shows that
such categories have enough injectives, due to a result of
Grothendieck which is given in the Grimoire as theorem 4.31.

May 14, 2021 : I've just completed the final reading of the first
section of Lion; it's a long section
that contains, among others, my treatment of faithfully flat descent,
which has undergone a thorough revision : it was rather hard to read
and some explanations were not very conceptual and rather clumsy;
now it's much better. Also, I've improved the presentation of the
basics of the Cech complex : previously the notation was unnecessarily
cumbersome and probably confusing, so also this part is now rather
better.

June 6, 2021 : The final reading of Lion
is now complete! That took almost exactly one month, which helps us
to put us back in schedule, but the road ahead is still long.
One notable small improvement is in the tratment of the glueing
of ringed spaces : now this is presented more explicitly as a
coequalizer of two glueing morphisms (lemma 5.74). Also, I've
added a little exercise, showing that the presheaves and sheaves
of modules over a sheaf of rings form a Grothendieck category with
a generator : this completes an exercise already mentioned in the
previous chapter; the proof actually reduces to that previous
exercise, via the technique of tensor products of presheaves
(and sheaves) of modules.